Kerodon

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Construction 6.1.2.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $T$, $C$, and $D$, together with $1$-morphisms $f: C \rightarrow D$, $g: D \rightarrow C$, $c: T \rightarrow C$, and $d: T \rightarrow D$.

  • Let $\epsilon : f \circ g \Rightarrow \operatorname{id}_ D$ and $\beta : x \Rightarrow g \circ d$ be $2$-morphisms of $\operatorname{\mathcal{C}}$. We will refer to the composition

    \[ f \circ c \xRightarrow {\operatorname{id}_ f \circ \beta } f \circ (g \circ d) \xRightarrow [\sim ]{\alpha _{f,g,d}} (f \circ g) \circ d \xRightarrow {\epsilon \circ \operatorname{id}_ d} \operatorname{id}_ D \circ d \xRightarrow [\sim ]{\lambda _ d} d \]

    as the left adjunct of $\beta $ with respect to $\epsilon $, or more simply as the left adjunct of $\beta $ if the $2$-morphism $\epsilon $ is clear from context. Here $\lambda _{d}$ and $\alpha _{f,g,d}$ are the left unit and associativity constraints for the $2$-category $\operatorname{\mathcal{C}}$.

  • Let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\gamma : f \circ c \Rightarrow d$ be $2$-morphisms of $\operatorname{\mathcal{C}}$. We will refer to the composition

    \[ c \xRightarrow [\sim ]{ \lambda _{c}^{-1} } \operatorname{id}_{C} \circ c \xRightarrow {\eta } (g \circ f) \circ c \xRightarrow [\sim ]{ \alpha _{g,f,c}^{-1} } g \circ (f \circ c) \xRightarrow {\operatorname{id}_ g \circ \gamma } g \circ d \]

    as the right adjunct of $\gamma $ with respect to $\eta $, or more simply as the right adjunct of $\gamma $ if the $2$-morphism $\eta $ is clear from context. Here again $\lambda _{c}$ and $\alpha _{g,f,c}$ are the left unit and associativity constraints for the $2$-category $\operatorname{\mathcal{C}}$.